3.1030 \(\int \frac {\coth (\sqrt {x}) \text {csch}(\sqrt {x})}{\sqrt {x}} \, dx\)

Optimal. Leaf size=8 \[ -2 \text {csch}\left (\sqrt {x}\right ) \]

[Out]

-2*csch(x^(1/2))

________________________________________________________________________________________

Rubi [A]  time = 0.19, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6715, 2606, 8} \[ -2 \text {csch}\left (\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(Coth[Sqrt[x]]*Csch[Sqrt[x]])/Sqrt[x],x]

[Out]

-2*Csch[Sqrt[x]]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps

\begin {align*} \int \frac {\coth \left (\sqrt {x}\right ) \text {csch}\left (\sqrt {x}\right )}{\sqrt {x}} \, dx &=2 \operatorname {Subst}\left (\int \coth (x) \text {csch}(x) \, dx,x,\sqrt {x}\right )\\ &=-\left (2 i \operatorname {Subst}\left (\int 1 \, dx,x,-i \text {csch}\left (\sqrt {x}\right )\right )\right )\\ &=-2 \text {csch}\left (\sqrt {x}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 8, normalized size = 1.00 \[ -2 \text {csch}\left (\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Coth[Sqrt[x]]*Csch[Sqrt[x]])/Sqrt[x],x]

[Out]

-2*Csch[Sqrt[x]]

________________________________________________________________________________________

fricas [B]  time = 0.47, size = 37, normalized size = 4.62 \[ -\frac {4 \, {\left (\cosh \left (\sqrt {x}\right ) + \sinh \left (\sqrt {x}\right )\right )}}{\cosh \left (\sqrt {x}\right )^{2} + 2 \, \cosh \left (\sqrt {x}\right ) \sinh \left (\sqrt {x}\right ) + \sinh \left (\sqrt {x}\right )^{2} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x^(1/2))*csch(x^(1/2))/x^(1/2),x, algorithm="fricas")

[Out]

-4*(cosh(sqrt(x)) + sinh(sqrt(x)))/(cosh(sqrt(x))^2 + 2*cosh(sqrt(x))*sinh(sqrt(x)) + sinh(sqrt(x))^2 - 1)

________________________________________________________________________________________

giac [B]  time = 0.13, size = 17, normalized size = 2.12 \[ \frac {4}{e^{\left (-\sqrt {x}\right )} - e^{\sqrt {x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x^(1/2))*csch(x^(1/2))/x^(1/2),x, algorithm="giac")

[Out]

4/(e^(-sqrt(x)) - e^sqrt(x))

________________________________________________________________________________________

maple [A]  time = 0.16, size = 7, normalized size = 0.88 \[ -2 \,\mathrm {csch}\left (\sqrt {x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x^(1/2))*csch(x^(1/2))/x^(1/2),x)

[Out]

-2*csch(x^(1/2))

________________________________________________________________________________________

maxima [B]  time = 0.31, size = 17, normalized size = 2.12 \[ \frac {4}{e^{\left (-\sqrt {x}\right )} - e^{\sqrt {x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x^(1/2))*csch(x^(1/2))/x^(1/2),x, algorithm="maxima")

[Out]

4/(e^(-sqrt(x)) - e^sqrt(x))

________________________________________________________________________________________

mupad [B]  time = 1.75, size = 8, normalized size = 1.00 \[ -\frac {2}{\mathrm {sinh}\left (\sqrt {x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x^(1/2))/(x^(1/2)*sinh(x^(1/2))),x)

[Out]

-2/sinh(x^(1/2))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth {\left (\sqrt {x} \right )} \operatorname {csch}{\left (\sqrt {x} \right )}}{\sqrt {x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x**(1/2))*csch(x**(1/2))/x**(1/2),x)

[Out]

Integral(coth(sqrt(x))*csch(sqrt(x))/sqrt(x), x)

________________________________________________________________________________________